#### Transcript Ch2 - University of Central Oklahoma

Signals and Systems Chapter 2 Biomedical Engineering Dr. Mohamed Bingabr University of Central Oklahoma Outline • Signals • Systems • The Fourier Transform • Properties of the Fourier Transform • Transfer Function • Circular Symmetry and the Hankel Transform Introduction Signal Type - Continuous Signal: x-ray attenuation - Discrete Signal: times of arrival of photons in a radioactive decay process in PET - Mixed signal: CT scan signal g(l,θk) System Type - Continuous-continuous system - Continuous-discrete system Signals 2-D continuous signal is defined as f(x,y) (x,y) : is a pixel location f : is pixel intensity function image Point Impulse 1-D point impulse (delta, Dirac, impulse function) 𝛿 𝑥 = 0, 𝑥 ≠ 0, 𝛿 𝑥 ∞ 𝑓(𝑥) 𝛿 𝑥 𝑑𝑥 = 𝑓 0 . −∞ 2-D point impulse 𝛿 𝑥, 𝑦 = 0, ∞ (𝑥, 𝑦) ≠ (0, 0) ∞ 𝑓(𝑥, 𝑦) 𝛿 𝑥, 𝑦 𝑑𝑥𝑑𝑦 = 𝑓 0, 0 . −∞ −∞ Point impulse is used in the characterization of image resolution and sampling 𝑥 Point Impulse Properties 1- Sifting property ∞ ∞ 𝑓(𝑥, 𝑦) 𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 𝑑𝑥𝑑𝑦 = 𝑓 𝜉, 𝜂 . −∞ −∞ We can interpret the product of a function with a point impulse as another point impulse whose volume is equal to the value of the function at the location of the point impulse. 2- Scaling property 1 𝛿 𝑎𝑥, 𝑏𝑦 = 𝛿(𝑥, 𝑦) 𝑎𝑏 2- Even function 𝛿 −𝑥, −𝑦 = 𝛿(𝑥, 𝑦) Line Impulse Line also used to assist image resolution 𝐿 𝑙, 𝜃 = 𝑥, 𝑦 |𝑥𝑐𝑜𝑠𝜃 + 𝑦𝑠𝑖𝑛𝜃 = 𝑙 This is a line whose unite normal is oriented at an angle θ relative to the x-axis and is at distance l from the origin in the direction of the unit normal. The line impulse 𝛿𝑙 𝑥, 𝑦 associated with line 𝐿 𝑙, 𝜃 𝛿𝑙 𝑥, 𝑦 = 𝛿𝑙 𝑥𝑐𝑜𝑠𝜃, +𝑦𝑠𝑖𝑛𝜃 − 𝑙 Comb and Sampling Functions Used in medical imaging production (sampling CT image 1024 x 1024), manipulation, and storage. ∞ 𝑐𝑜𝑚𝑏(𝑥) = 𝛿(𝑥 − 𝑛) −∞ 2-D comb function ∞ ∞ 𝑐𝑜𝑚𝑏(𝑥, 𝑦) = 𝛿(𝑥 − 𝑚, 𝑦 − 𝑛) 𝑚=−∞ 𝑛=−∞ Sampling function ∞ ∞ 𝛿𝑠 𝑥, 𝑦; ∆𝑥, ∆𝑦 = 𝛿(𝑥 − 𝑚∆𝑥, 𝑦 − 𝑛∆𝑦) 𝑚=−∞ 𝑛=−∞ 1-D Rect and Sinc Functions Rect function is used in medical imaging for sectioning. 𝑟𝑒𝑐𝑡(𝑥) = 1, 0, 1 for |𝑥| < 2 1 for |𝑥| > 2 Sinc function is used in medical imaging for reconstruction. 𝑠𝑖𝑛𝜋𝑥 𝑠𝑖𝑛𝑐 𝑥 = 𝜋𝑥 2-D Rect and Sinc Functions 𝑟𝑒𝑐𝑡(𝑥, 𝑦) = 𝑟𝑒𝑐𝑡(𝑥)𝑟𝑒𝑐𝑡(𝑦) 𝑟𝑒𝑐𝑡(𝑥, 𝑦) = 1, 0, 1 1 for |𝑥| < and |𝑦| < 2 2 1 1 for |𝑥| > and 𝑦 > 2 2 𝑠𝑖𝑛𝑐 𝑥, 𝑦 = 𝑠𝑖𝑛𝑐 𝑥 𝑠𝑖𝑛𝑐(𝑦) 1, 𝑠𝑖𝑛𝑐(𝑥, 𝑦) = sin 𝜋𝑥 sin(𝜋𝑦) , 2 𝜋 𝑥𝑦 for 𝑥 = 𝑦 = 0 otherwise. Exponential and Sinusoidal Signals 𝑒(𝑥, 𝑦) = 𝑒 𝑗2𝜋 𝑢0 𝑥+𝑣0 𝑦 𝑒 𝑥, 𝑦 = 𝑐𝑜𝑠 2𝜋 𝑢0 𝑥 + 𝑣0 𝑦 𝑐𝑜𝑠 2𝜋 𝑢0 𝑥 + 𝑣0 𝑦 = 0.5𝑒 𝑗2𝜋 x and y have distance units. u0 and v0 are the fundamental frequencies and their units are the inverse of the units of x and y. + 𝑗𝑠𝑖𝑛 2𝜋 𝑢0 𝑥 + 𝑣0 𝑦 𝑢0 𝑥+𝑣0 𝑦 + 0.5𝑒 −𝑗2𝜋 𝑢0 𝑥+𝑣0 𝑦 Separable and Periodic Signals • A signal f(x, y) is separable if f(x, y)= f1(x) f2(y) • Separable signal model signal variations independently in the x and y direction. • Decomposing a signal to its components f1(x) and f2(y) might simplify signal processing. Periodicity A signal f(x, y) is periodic if f(x, y)= f(x+X, y) = f(x, y+Y) X and Y are the signal periods in the x and y direction, respectively. Systems A continuous system is defined as a transformer Ϩ of an input continuous signal f(x,y) to an output continuous signal g(x,y). g(x, y)= Ϩ [f(x, y)] Linear Systems 𝐾 𝐾 𝑤𝑘 𝑓𝑘 (𝑥, 𝑦) = Ϩ 𝑘=1 𝑤𝑘 Ϩ 𝑓𝑘 (𝑥, 𝑦) 𝑘=1 Impulse Response If we know the system response to an impulse 𝛿𝜉𝜂 𝑥, 𝑦 = 𝛿 𝑥 − 𝜉, 𝑦 − 𝜂 then with linearity we can know the system response to any input. ℎ 𝑥, 𝑦; 𝜉, 𝜂 = Ϩ 𝛿𝜉𝜂 𝑥, 𝑦 ℎ 𝑥, 𝑦; 𝜉, 𝜂 is the system impulse response function or known as point spread function (PSF). Impulse Response System output g() for any input f(). ∞ 𝑔 𝑥, 𝑦 = ∞ 𝑓 𝜉, 𝜂 ℎ 𝑥, 𝑦; 𝜉, 𝜂 𝑑𝜉𝑑𝜂 −∞ −∞ Shift Invariance System A system is shift invariant if an arbitrary translation of the input results in an identical translation in the output. Then with linearity we can know the system response to any input. Let the input 𝑓𝑥0 𝑦0 𝑥, 𝑦 = 𝑓 𝑥 − 𝑥0 , 𝑦 − 𝑦0 then the output g(𝑥 − 𝑥0 , 𝑦 − 𝑦0 )= Ϩ [𝑓𝑥0 𝑦0 𝑥, 𝑦 ] System response to a shifted impulse Ϩ [𝛿𝜉𝜂 𝑥, 𝑦 ]= h(𝑥 − 𝜉, 𝑦 − 𝜂) Linear Shift-Invariance (LSI) System Linear shift-invariant (LSI) System Response ∞ ∞ 𝑔 𝑥, 𝑦 = 𝑓 𝜉, 𝜂 ℎ 𝑥 − 𝜉, 𝑦 − 𝜂 𝑑𝜉𝑑𝜂 −∞ −∞ Convolution Integral representation of system response 𝑔 𝑥, 𝑦 = ℎ 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦) Example: Consider a continuous system with inputoutput equation g(x,y) = xyf(x,y). Is the system linear and shift-invariant? Connection of LSI Systems Cascade Parallel 𝑔 𝑥, 𝑦 = ℎ1 𝑥, 𝑦 ∗ ℎ2 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦) 𝑔 𝑥, 𝑦 = [ℎ1 𝑥, 𝑦 + ℎ2 𝑥, 𝑦 ] ∗ 𝑓(𝑥, 𝑦) Connection of LSI Systems Example: Consider two LSI systems connected in cascade, with Gaussian PSFs of the form: 1 − ℎ1 𝑥, 𝑦 = 𝑒 2𝜋𝜎12 𝑥 2 +𝑦 2 /2𝜎12 1 − ℎ2 𝑥, 𝑦 = 𝑒 2𝜋𝜎22 𝑥 2 +𝑦 2 /2𝜎22 where σ1 and σ2 are two positive constants. What is the PSF of the system? Separable Systems A 2-D LSI system with PSF h(x, y) is a separable system if there are two 1-D systems with PSFs h1(x) and h2(x), such that h(x,y) = h1(x)h2(x) 1 − ℎ 𝑥, 𝑦 = 𝑒 2𝜋𝜎 2 𝑥 2 +𝑦 2 /2𝜎 2 This PSF is separable 1 2 /2𝜎 2 −𝑥 ℎ1 𝑥 = 𝑒 2𝜋𝜎 1 2 /2𝜎 2 −𝑦 ℎ2 𝑥 = 𝑒 2𝜋𝜎 Separable Systems In practice it is easier and faster to execute two consecutive 1-D operations than a single 2-D operation. ∞ 𝑤 𝑥, 𝑦 = 𝑓 𝜉, 𝑦 ℎ1 𝑥 − 𝜉 𝑑𝜉 −∞ g 𝑥, 𝑦 = For every y ∞ 𝑤 𝑥, 𝜂 ℎ2 𝑦 − 𝜂 𝑑𝜂 −∞ For every x Stable Systems A system is a bounded-input bounded-output (BIBO) stable system if For bounded input |𝑓 𝑥, 𝑦 | ≤ 𝐵 < ∞ for every (x, y) The output is bounded 𝑔(𝑥, 𝑦) = ℎ 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦) ≤ 𝐵, < ∞ and ∞ ∞ |ℎ 𝑥, 𝑦 |𝑑𝑥𝑑𝑦 < ∞ −∞ −∞ 1-D Fourier Transform (time) Continuous 1-D Fourier Transform j 2ft x ( t ) e dt X (2f ) Discrete 1-D Fourier Transform N -1 X ( k ) x ( n )e j 2k n N n 0 x(n) X(k) = [125 = [668 |X(k)| = [668 Phase = [0 145 -29.2 - j38 47.9 -127.5 148 7.7 - j12.96 15.1 -59.3 140 7.7 - j12.96 15.1 59.3 110] -29.2 - j38] 47.9] 127.5] 1-D Fourier Transform 1-D Fourier transform ∞ 𝐹 𝑢 = ℱ1𝐷 𝑓 𝑢 = 𝑓 𝑥 𝑒 −𝑗2𝜋𝑢𝑥 𝑑𝑥 −∞ u is the spatial frequency 1-D inverse Fourier transform −1 𝑓 𝑥 = ℱ1𝐷 𝐹 𝑥 = Example: ∞ 𝐹 𝑢 𝑒 𝑗2𝜋𝑢𝑥 𝑑𝑢 −∞ What is the Fourier 1, transform of the 𝑟𝑒𝑐𝑡(𝑥) = 0, 1 for |𝑥| < 2 1 for |𝑥| > 2 Fourier Transform The 2-D Fourier transform of f(x, y) ∞ 𝐹 𝑢, 𝑣 = ℱ2𝐷 𝑓 𝑢, 𝑣 = ∞ 𝑓 𝑥, 𝑦 𝑒 −𝑗2𝜋(𝑢𝑥+𝑣𝑦) 𝑑𝑥𝑑𝑦 −∞ −∞ u and v are the spatial frequencies The 2-D inverse Fourier transform of F(u, v) ∞ 𝑓(𝑥, 𝑦) = ∞ 𝐹 𝑢, 𝑣 𝑒 𝑗2𝜋(𝑢𝑥+𝑣𝑦) 𝑑𝑢𝑑𝑣 −∞ −∞ Fourier Transform Magnitude (magnitude spectrum) of FT 𝐹(𝑢, 𝑣) = 𝐹𝑅2 𝑢, 𝑣 + 𝐹𝐼2 𝑢, 𝑣 Angle (phase spectrum) of the FT ∠𝐹 𝑢, 𝑣 = 𝑡𝑎𝑛 −1 𝐹𝐼 (𝑢, 𝑣) 𝐹𝑅 (𝑢, 𝑣) 𝐹(𝑢, 𝑣) = 𝐹(𝑢, 𝑣) 𝑒 𝑗∠𝐹 𝑢,𝑣 Example: What is the Fourier transform of the point impulse 𝛿(𝑥, 𝑦)? Fourier Transform Pairs Examples of Fourier Transform Example: What is the Fourier transform of 𝑓 𝑥, 𝑦 = 𝑒 𝑗2𝜋(𝑢0 𝑥+𝑣0𝑦) Answer: ℱ2𝐷 𝑓 𝑢, 𝑣 = 𝛿 𝑢 − 𝑢0 , 𝑣 − 𝑣0 If the spatial frequency u0 and v0 are zero then f(x,y) =1 and the spectrum F(u,v) will be 𝛿 𝑢, 𝑣 . Slow signal variation in space produces a spectral content that is primarily concentrated at low frequencies. Examples of Fourier Transform Three images of decreasing spatial variation (from left to right) and the associated magnitude spectra [depicted as log(1 + |F(u, υ)|)]. Examples of Fourier Transform >> img1 = imread('\\PHYSICSSERVER\MBingabr\BiomedicalImaging\mri.tif'); >> imshow(img1) >> size(img1) ans = 256 256 >> FFT_img1 = fftshift(fft2(img1)); >> Abs_FFT_img1 = abs(FFT_img1) >> surf(Abs_FFT_img1(110:140,110:140)) >> Log_Abs_FFT_img1=log10(1+Abs_FFT_img1); >> surf(Log_Abs_FFT_img1(110:140,110:140)) Properties of the Fourier Transform Properties are used in theory and application to simplify calculation. Linearity ℱ2𝐷 𝑎1 𝑓 + 𝑎2 𝑔 𝑢, 𝑣 = 𝑎1 𝐹 𝑢, 𝑣 + 𝑎2 𝐺(𝑢, 𝑣) Translation If F(u,v) is the FT of a signal f(x, y) then the FT of a translated signal 𝑓𝑥0 𝑦0 𝑥, 𝑦 = 𝑓 𝑥 − 𝑥0 , 𝑦 − 𝑦0 is ℱ2𝐷 𝑓𝑥0 𝑦0 𝑢, 𝑣 = 𝐹 𝑢, 𝑣 𝑒 −𝑗2𝜋(𝑢𝑥0 +𝑣𝑦0) Properties of the Fourier Transform Conjugation and Conjugate Symmetry If F(u,v) is the FT of a signal f(x, y) then 𝐹(𝑢, 𝑣) = 𝐹 ∗ (−𝑢, −𝑣) 𝐹𝑅 (𝑢, 𝑣) = 𝐹𝑅 (−𝑢, −𝑣) |𝐹 𝑢, 𝑣 | = |𝐹 −𝑢, −𝑣 | ∠𝐹 𝑢, 𝑣 = −∠𝐹 −𝑢, −𝑣 𝐹𝐼 (𝑢, 𝑣) = −𝐹𝐼 (−𝑢, −𝑣) Properties of the Fourier Transform Scaling If F(u,v) is the FT of a signal f(x, y) and if 𝑓𝑎𝑏 𝑥, 𝑦 = 𝑓 𝑎𝑥, 𝑏𝑦 1 𝑢 𝑣 ℱ2𝐷 𝑓𝑎𝑏 𝑢, 𝑣 = 𝐹 , |𝑎𝑏| 𝑎 𝑏 Example Detectors of many medical imaging systems can be modeled as rect functions of different sizes and locations. Compute the FT of the following 𝑥 − 𝑥0 𝑦 − 𝑦0 𝑓 𝑥, 𝑦 = 𝑟𝑒𝑐𝑡 , ∆𝑥 ∆𝑦 Properties of the Fourier Transform Rotation If F(u,v) is the FT of a signal f(x, y) and if 𝑓𝜃 𝑥, 𝑦 = 𝑓 𝑥𝑐𝑜𝑠𝜃 − 𝑦𝑠𝑖𝑛𝜃, 𝑥𝑠𝑖𝑛𝜃 + 𝑦𝑐𝑜𝑠𝜃 ℱ2𝐷 𝑓𝜃 𝑢, 𝑣 = 𝐹 𝑢 𝑐𝑜𝑠𝜃 − 𝑣 𝑠𝑖𝑛𝜃, 𝑢 𝑠𝑖𝑛𝜃 + 𝑣 𝑐𝑜𝑠𝜃 If f(x, y) is rotated by an angle 𝜃, then its FT is rotated by the same angle. Properties of the Fourier Transform Convolution The Fourier transform of the convolution f(x, y) * g(x, y) is ℱ2𝐷 𝑓 ∗ 𝑔 𝑢, 𝑣 = 𝐹 𝑢 , 𝑣 𝐺 𝑢 , 𝑣 Convolution property simplify the difficult task of calculating the convolution in the spatial domain to multiplication in the frequency domain. Example: Find Fourier transform of the convolution f(x, y) * g(x, y) 𝑔 𝑥, 𝑦 = 𝑠𝑖𝑛𝑐 𝑉𝑥, 𝑈𝑦 𝑓 𝑥, 𝑦 = 𝑠𝑖𝑛𝑐 𝑈𝑥, 𝑉𝑦 0<VU Properties of the Fourier Transform Product The Fourier transform of the product f(x, y) g(x, y) is the convolution of their Fourier transforms. ℱ2𝐷 𝑓𝑔 𝑢, 𝑣 = 𝐹 𝑢 , 𝑣 ∗ 𝐺 𝑢 , 𝑣 ∞ = ∞ 𝐺 𝜉, 𝜂 𝐹 𝑢 − 𝜉, 𝑣 − 𝜂 𝑑𝜉𝑑𝜂 −∞ −∞ Separable Product If f(x, y)=f1(x)f2(y) then ℱ2𝐷 𝑓 𝑢, 𝑣 = 𝐹1 (𝑢)𝐹2 (𝑣) where 𝐹1 𝑢 = ℱ1𝐷 𝑓1 𝑢 Separability of the Fourier Transform The Fourier transform F(u,v) of a 2-D signal f(x, y) can be calculated using two simpler 1-D Fourier transforms, as follows: ∞ 1) 𝑟 𝑢, 𝑦 = 𝑓 𝑥, 𝑦 𝑒 −𝑗2𝜋𝑢𝑥 𝑑𝑥 −∞ ∞ 2) 𝐹 𝑢, 𝑦 = −∞ 𝑟 𝑢, 𝑦 𝑒 −𝑗2𝜋𝑢𝑦 𝑑𝑦 For every y. For every x. Transfer Function The system’s transfer function (frequency response) H(u, v) is the Fourier transform of the system’s PSF h(x,y). ∞ ∞ 𝐻 𝑢, 𝑣 = ℎ 𝜉, 𝜂 𝑒 −𝑗2𝜋(𝑢𝜉+𝑣𝜂) 𝑑𝜉𝑑𝜂 −∞ −∞ The inverse Fourier transform of the transfer function H(u, v) is the point spread function h(x,y). ∞ ∞ ℎ 𝑥, 𝑦 = 𝐻 𝑢, 𝑣 𝑒 −𝑗2𝜋(𝑢𝑥+𝑣𝑦) 𝑑𝑢𝑑𝑣 −∞ −∞ The output G(u, v) of a system in response to input F(u, v) is the product of the input with the transfer function H(u, v) . 𝐺 𝑢, 𝑣 = 𝐻 𝑢 , 𝑣 𝐹 𝑢 , 𝑣 Transfer Function Example: Consider an idealized system whose PSF is h(x,y) = (x-x0, y-y0). What is the transfer function H(u, v) of the system, and what is the system output g(x,y) to an input signal f(x,y). Low Pass Filter 𝐻(𝑢, 𝑣) = 𝐺(𝑢, 𝑣) = 1, for 𝑢2 + 𝑣 2 ≤ 𝑐 0, for 𝑢2 + 𝑣 2 > 𝑐 𝐹(𝑢, 𝑣) for 𝑢2 + 𝑣 2 ≤ 𝑐 0, for 𝑢2 + 𝑣 2 > 𝑐 c1 > c2 Circular Symmetry Often, the performance of a medical imaging system does not depend on the orientation of the patient with respect to the system. The independence arises from the circular symmetry of the PSF. A 2-D signal f(x, y) is circularly symmetric if fθ(x, y) = f(x, y) for every θ. Property of Circular Symmetry • f(x, y) is even in both x and y • F(u, v) is even in both u and v • | F(u, v) | = F(u, v) • ∠F(u, v) = 0 • f(x, y) = f(r) where 𝑟 = 𝑥2 + 𝑦2 • F(u, v) = F(q) where 𝑞 = 𝑢2 + 𝑣 2 f(r) and F(q) are one dimensional signals representing two dimensional signals Hankel Transform The relationship between f(r) and F(q) is determined by Hankel Transform. ∞ 𝐹 𝑞 = 2𝜋 0 𝑓 𝑟 𝐽𝑜 2𝜋𝑞𝑟 𝑟𝑑𝑟 𝐹 𝑞 = ℋ 𝑓(𝑟) where J0(r) is the zero-order Bessel function of the first kind. 1 𝜋 𝐽0 𝑟 = cos 𝑟𝑠𝑖𝑛𝜙 𝑑𝜙 𝜋 0 The nth-order Bessel function 1 𝜋 𝐽𝑛 𝑟 = cos 𝑛𝜙 − 𝑟𝑠𝑖𝑛𝜙 𝑑𝜙 𝜋 0 for n = 0, 1, 2, … Hankel Transform The inverse Hankel transform. ∞ 𝑓 𝑟 = 2𝜋 0 unit disk 𝐹 𝑞 𝐽𝑜 2𝜋𝑞𝑟 𝑞𝑑𝑞 jink function sinc and jink functions Example In some medical imaging systems, only spatial frequencies smaller than q0 can be imaged. What is the function having uniform spatial frequencies within the desk of radius q0 and what is its inverse Fourier transform.